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MODEL OF ULTRASONIC RAY PATHS IN AN ANISOTROPIC WELD BASED ON RAY METHOD Q. Liu, H. Wirdelius Chalmers Lindholmen University College, Göteborg, Sweden Abstract: Experiences from isotropic materials suggest that it is impossible to derive a statistical model with few parameters to model a weld in an anisotropic material. The parameter that remains unknown even after a large number of empirical investigations is the orientation of the dendrites due to the heating. Our intention is to apply some simplified models in an optimization scheme to predict the dendrites' orientation in different kinds of welds. In this paper, a model of ultrasonic ray traveling through anisotropic materials, as well as the corresponding computer simulation is presented. First, based on the previous work on texture analysis of various austenitic stainless steel weld samples, a typical simplified model of the geometry of welds is established. It is subdivided into a finite number of smaller regions with specific anisotropy. Then by means of the ray tracing method, ray path running through the weld model is developed. As a function of incident angle, the point of entrance of the sound path to the weld and the given orientation of the dendrites, the model will ultimately give the point where the sound leaves the weld, the time of flight and the angle between the sound ray and the weld. A numerical simulation of the forward problem is also provided as a validation of the model. Introduction: Non destructive testing (NDT) is a commonly used industrial method to evaluate the integrity of individual components. In-service induced cracks such as fatigue and stress corrosion cracks can, if they are detected, be sized and monitored in order to postpone repairs or replacements. The reliability of a NDT method is highly dependent on how the equipment is adjusted to a specific object and to anticipated crack features. If ultrasonic NDT is applied on materials with strong anisotropy (e.g. welds in stainless steel) it can introduce possible difficulties in the interpretation as the ultrasound tends to bend in an unpredicted way. Ultrasonic NDT techniques to size defects are mainly based on the information of ultrasound paths from the crack tips and thus very sensitive to these kind of effects. According to the Swedish Nuclear Power Inspectorate's requirements in the regulations concerning structural components in nuclear installations, in-service inspection must be performed using inspection methods that have been qualified. These demands on reliability of used NDE/NDT procedures and methods have stimulated the development of simulation tools of NDT. Experiences done by modeling and verification towards isotropic materials suggest that it is not possible to derive a statistical model with few parameters to model a weld in an anisotropic matrix material. The parameter that remains unknown even after a large number of empirical investigations is the orientation of the dendrites due to the heating. This is also probably the most important parameter in determining the sound path through the weld. Numerical methods based on discretization of the simulated volume (e.g. finite element methods and EFIT [1]) are well suited to model objects with complex material structure. Since the number of elements to ensure accuracy in this kind of methods soon increases to an unmanageable amount, simulation has in general been limited to two-dimensional configurations ([2], [3]). Exact methods have been applied to less compound structures (layered anisotropy), though full three-dimensional objects including crack scattering ([4]). The solidification process is individual for each weld and the resulting dendrite orientation (and anisotropy) can only be assumed to be homogenous in small fractions of the volume. It is therefore difficult to find simple functions to describe the distribution of the orientation. Ray tracing techniques based on high frequency approximations have been applied to model three-dimensional welded volumes ([5], [6], [7] and [8]). Often these utilize subdivision of the volume in homogenous parts or descriptions of the orientation by functions based on ad hoc assumptions of the solidification process. The present paper gives a short description of a simple ray tracing model of ultrasonic propagation through a welded region. This paper is the first part in a project that will develop an ultrasonic technique to retrieve the anisotropy directivity in a welded region. This model together with experimental data is to be implemented in an optimization algorithm to calculate the orientation of the dendrites in the real weld. The final application has enforced the simplicity and reduced the number of possible parameters within the model. The paper initially starts of with a declaration and justification of the model being limited to represent transversely isotropy material. Thereafter a simplified model of the geometry of welds is deduced. This is based on the previous work on texture analysis of various austenitic stainless steel weld samples and also subdivides the volume in a finite number of smaller regions with homogeneous anisotropy. The ray tracing model is subsequently presented and the paper also includes a minor numerical example. Elastic anisotropy in stainless steel weld: A material can be stated to be anisotropic if its properties, when measured at the same location, change with direction. The simplest anisotropic case of broad engineering applicability is transversely isotropy, which has one distinct direction, while the other two directions are equivalent to each other which form an isotropic plane. Earlier works on both the application of ultrasonic NDT and the destructive examination of welds have shown that the austenitic stainless steel weld can be modeled as transversely isotropy, due to the ordered columnar structure, similar to a fiber texture. That is to say, around some principal direction, there is a random angular orientation of the crystals. Thus, in a plane perpendicular to these preferred directions the material is assumed to be isotropic. Due to this kind of structural symmetry, the elastic modulus matrix thus has five independent components among twelve nonzero components, giving the elastic modulus matrix the form ([9]) 11 12 13 12 11 13 13 13 33 44 c c c c c c c c c C c = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ) c =- 2 c c and the third direction is taken as the principal axis. Model of the geometry of a typical weld: The extensive columnar grain structure in austenitic welds differs greatly from that in ferritic welds. In both cases, the solidification process during welding initially produces a columnar grain structure in each weld bead. Grains grow along the maximum thermal gradients in the bead ([10]). Growth in one particular direction is faster than in other directions and this leads to the rapid disappearance of out-of-plane oriented grains. In the following, the deposition of subsequent weld metal reheats the beads partially and, in the case of a ferritic weld, the columnar grain structure is destroyed by the austenite-ferrite phase transformation that occurs as the solid cools. No such transition occurs in the austenitic alloys and consequently the columnar grain structure survives. Furthermore, each new weld bead re-melts the surface of the preceding beads and the new grains grow epitaxially on the existing ones. Consequently grains of substantial length are produced. In Fig.1, a macrograph of a typical V-butt weld is shown. 44 c 66 c (1) where, 66 11 12 X2 X1 X3 Fig.1 Macrograph of an austenitic stainless steel weld Seen from the picture, apparently, the crystalline structure is composed of a series of long grains, whose dendrite directions are spatially varied. The columnar structure of the austenitic weld shows that the long columnar dendrites begins at the fusion lines and extends into the body of the weld. What's more, the long dendrite axis is almost vertical along the centre of the weld and nearly perpendicular to the fusion lines and the upper boundary of the weld. This presents the possibility of modeling the dendrite directions, in a primitive assumption of the structure, with the following expression (2 3 1 1 1 iX a X X i i i bi =- )a i (2) where, are defined by the geometric relation at the fusion line and upper boundary. The values of these coefficients may be changed according to certain assumption made to the structure of a specific weld. 1 2 1 a a X i, , i b X1 X1 X3 X3 Fig.2 Plot of the weld model Fig.3 Plot of the sub-regions in a weld model Equation (2) is utilized to define the boundaries of the dendrite orientations in Fig. 1 and an example of this is visualized in Fig. 2. The regions outside are modeled as ferritic materials and are assumed to have homogeneous and isotropic elasticity properties (almost certainly true compared with the welded region). According to the requirement of modeling and simulation, the whole weld is thereafter subdivided into smaller regions with predefined homogeneous transversely anisotropy (Fig. 3). The directivity in each fraction is given by the mean value of that provided by the centre of its boundaries. The elastic property in each fraction is assumed almost the same and then can be regarded as homogeneous transversely isotropy. Ray tracing model: In order to simulate an ultrasound wave propagating through a weld, a mathematical model is established here. In this paper, we only consider the propagating direction of a quasi P-wave (also called longitudinal wave). No information about the amplitude of rays is deduced and the analysis is limited to the quasi P-wave mode. The model comprises mainly the following parts. 1. Expression of the phase velocity as a function of the phase angle Unlike the wave propagation in isotropic medium, the phase velocity of quasi P-wave in anisotropic material on the other hand is a function of the angle between the wave normal direction and the long axis of the columnar grains. For a general elastic anisotropic medium, in the absence of body forces, the wave equation may be written in the form ([11]), |
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